A073918 Smallest prime which is 1 more than a product of n distinct primes: a(n) is a prime and a(n) - 1 is a squarefree number with n prime factors.

2, 3, 7, 31, 211, 2311, 43891, 870871, 13123111, 300690391, 6915878971, 200560490131, 11406069164491, 386480064480511, 18826412648012971, 693386350578511591, 37508276737897976011, 3087649419126112110271, 183452981525059000664911, 11465419967969569966774411

Offset: 0

Author

Amarnath Murthy, Aug 18 2002

Keywords

• nonn

Comments

Apparently the same as record values of A055734: least k such that phi(k) has n distinct prime factors, where phi is Euler's totient function. If the Mathematica program is used for large n, then "fact" should be reduced to, say, 1.1 in order to increase the search speed. - T. D. Noe, Dec 17 2003

Examples

			a(0) = 1 + 1 = 2 (empty product of zero primes).
a(1) = 1 + 2 = 3.
a(2) = 1 + 2*3 = 7.
a(3) = 1 + 2*3*5 = 31.
a(4) = 1 + 2*3*5*7 = 211.
a(5) = 1 + 2*3*5*7*11 = 1 + 11# = 2311.
a(6) = 1 + 2*3*5*7*11*19 = 43891, since 13# + 1 and 11#*17 + 1 = 17#/13 + 1 is not prime, and 17#/p + 1 is larger than a(6) for all p in {2, ..., 11}.
The index of the smallest prime which is not a factor of a(n)+1 is (1, 2, 3, 4, 5, 6, 6, 7, 7, 9, 10, 12, 11, 12, 13, 15, 16, 15, 16, 18, 19, 20, 21, 22, 22, 23, 25, 27, 26, 29, 29, ...) for n = 0, 1, 2, ... - _M. F. Hasler_, May 31 2018
		

Links

• Max Alekseyev, Table of n, a(n) for n = 0..100 (terms for n = 0..24 from M. F. Hasler)

Crossrefs

Cf. A055734 (number of distinct prime factors of phi(n)).

Cf. A073917, A098026.

Cf. A000040 (primes), A002110 (primorial), A081545 (same with composite instead of primes).

Programs

Mathematica

Generate[pIndex_, i_] := Module[{p2, t}, p2=pIndex; While[p2[[i]]++; Do[p2[[j]]=p2[[i]]+j-i, {j, i+1, Length[p2]}]; t=Times@@Prime[p2]; t
				

PARI

A073918(n, b=0 /*best*/, p=1 /*product*/, f=[]/*factors*/)={ if( #f= f[n+1] ) || !b = A073918( n-1, b, p*f[n], f), f[n]= nextprime( f[n]+1 ) ); b } \\ then, e.g.: apply(A073918, [0..30]). - M. F. Hasler Jun 16 2007

Formula

From M. F. Hasler, Jun 16 2007 (Start):

Conjecture: For any m > 0 there is K > 0 such that for all k > K, a(k)-1 is divisible by the first m primes.

Corollary: For any m > 1 there is K > 0 such that for all k > K, a(k) = 1 (mod m).

Conjecture 2: Let K(m) be the smallest possible K satisfying the above Conjecture. Then K(m) ~ m, i.e., a(k) ~ A002110(k), only very few of the last factors will be a bit larger. (End)

Remark: the last "~" above was not intended to mean asymptotic equivalence. It appears that lim inf a(n)/A002110(n) = 1, but the lim sup might well be larger. It would be interesting to know whether it has a finite value. - M. F. Hasler, May 31 2018

Extensions

More terms from Vladeta Jovovic, Aug 20 2002

Edited by M. F. Hasler, May 31 2018